CHAPTER 1 MATHEMATICAL CONCEPTS
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1 CHAPTER 1 MATHEMATICAL CONCEPTS Part I Expressing Numbers that are Very Large or Very Small 1. Scientific Notation In the study of chemistry we often encounter numbers that are very large or very small. It is more convenient to work with these numbers if we express them in scientific notation. This means that we move the decimal point to a position immediately to the right of the first non-zero digit and use a power of ten to indicate how many places the decimal point has been moved. Thus any number expressed in scientific notation has the following format: an exponent that indicates how far first digit the decimal point was moved decimal point x 10 rest of the digits Remember that the decimal point must always end up immediately to the right of the first digit in scientific notation. Let's look at some numbers and see how they are converted to scientific notation. regular form 3,500, , scientific notation 3.5 x x x x x 10-3 Notice that In scientific notation the first part is always a number between one and ten. Notice that In scientific notation the exponent of the power of ten indicates how many places the decimal point has been moved. Notice that If the original number was large (greater than one), the exponent is positive. If the original number was small (less than one). the exponent is negative. In the last example above the decimal point is moved 3 places Chap1 Math Concepts-JC.docx 05/12/11
2 Problems Copy each problem into your spiral notebook and convert each number into scientific notation , , Copy each problem into your notebook. Then get rid of the scientific notation and express each as a regular number x x x x Multiplication When we multiply numbers in scientific notation, we multiply the first part of each number as usual and add the exponents of the power of ten. Look over the following sample problems to see how this is done. Sample Problem 1 Multiply: Solution 1. Multiply 2x3 =6 2. Add 4+5=9 the answer 6x10 9 Sample Problem 2 Multiply: Solution 1. Multiply 4x2=8 2. Add The answer 8x Division When we divide numbers in scientific notation, we divide the first part of each number as usual and subtract the exponents of the power of ten (the top exponent minus the bottom exponent). Look over the next sample problems to see how this is done. Sample Problem 3 Divide: Solution 1. Divide 2. Subtract the answer Chap1 Math Concepts-JC.docx 05/12/11 2
3 6. Addition and Subtraction When doing calculations involving measurements, the answer can only be as accurate as the least accurate measurement made. For addition and subtraction this can be expressed in the following rule: When Adding or Subtracting The answer must be rounded off to the same column (ones, tenths, hundredths, etc.) as the least precise measurement used in the calculation. Usually the least number of decimal places Sample Problem 7 Subtract: Solution Calculator answer Rounded to the correct precision (to the nearest tenth) Sample Problem 8 Add: 4, Solution Calculator answer 5,228.6 Rounded to the correct precision 5,200 (to the nearest hundred) Problems Copy each problem into your notebook, do the addition or subtraction indicated, and round your answer to the correct precision , Chap1 Math Concepts-JC.docx 05/12/11 3
4 7. Significant Figures The significant figures are the digits in a number which represent the accuracy of that number. All non-zero digits in a number are significant. But zeros may be just "place holders". The following two examples show the use of place holders in numbers..085 This number has an accuracy of two significant figures. In this number the "8" and "5" are measured digits and are therefore significant. The zero is just a place holder that shows the position of the decimal point; it is not a significant figure. 400 This number has an accuracy of one significant figure. Trailing zeros are often only place holders. In this number the zeros are there to show that the "4" is in the hundreds column. Since no decimal point is shown, the zeros have not been measured and are not significant. For the beginner, it is often difficult to decide which digits are significant and which are not. For this reason, it is best to follow strictly a set of rules for determining the number of significant figures in any number. Rules for Determining Significant Figures 1. All non-zero digits are significant. 2. Zeros to the left of non-zero digits are NEVER significant. 3. Zeros between non-zero digits are ALWAYS significant. 4. Zeros to the right of non-zero digits are significant ONLY if a decimal point is shown. *Notice that the terms left, between and right refer to the placement of the zeros in relationship with nonzero numbers NOT in relationship with the decimal point. All non-zero digits are always significant. The following examples illustrate the rules shown above as they apply to zeros: rule 2 rule 3 rule 4 Number sig figs number sig figs number sig figs , , , Chap1 Math Concepts-JC.docx 05/12/11 4
5 Problems Copy each problem into your notebook. Indicate the number of significant figures and list the rules (by number) that apply to each , Multiplication and Division , ,200 When multiplying of dividing numbers you must count the number of significant figures in each number and round off the answer to the same number of significant figures as the least accurate number. When Multiplying or Dividing The answer must be rounded off to the same number of significant figures as the least accurate measurement used in the calculation. Sample Problem 9 Multiply: (34.0) ( ) Solution Count significant figures 3 6 Calculator answer Rounded to the correct accuracy 3.13 (to 3 significant figures) Sample Problem 10 Divide: Solution Count significant figures 6 / 1 Calculator answer Rounded to the correct accuracy 80 (to 1 significant figure) Chap1 Math Concepts-JC.docx 05/12/11 5
6 Problems Copy each problem into your notebook. Label each number as to how many significant figures it contains. Write down your calculator answer and then the answer rounded to the correct accuracy Summary of Methods The most common mistake is to mix up the two methods we have learned for rounding. Here are the to rules we have learned: When Adding or Subtracting The answer must be rounded off to the same column (ones, tenths, hundredths, etc.) as the least precise measurement used in the calculation. When Multiplying or Dividing The answer must be rounded off to the same number of significant figures as the least accurate measurement used in the calculation. Mixed Problems Copy each problem into your notebook and show both your calculator answer and the final answer (rounded correctly). Be careful to use the right method for each problem Chap1 Math Concepts-JC.docx 05/12/11 6
7 Rounding Off When rounding off numbers, if what you drop off is greater than five, round up; if what you drop off is less than five, leave what remains alone. This rule will take care of 99% of all situations. But on the rare occasion in which the part you drop off is exactly five, round what remains to an even number. These three situations are illustrated below as each number is rounded to two significant figures. Round Up Leave Alone Round Even The special "round even" rule is used rarely since even a number such as will be rounded up to 29 (if rounded to two significant figures). Chap1 Math Concepts-JC.docx 05/12/11 7
8 This ruler illustrates how numbers that are exactly halfway round off to an even number. Number: Round to: Counting Numbers All of our discussion of accuracy and rounding has been directed to numbers that are measurements, as most of the numbers we deal with are. But occasionally we work with counting numbers. Numbers that have been arrived at by counting (rather than by measuring) are exact and have unlimited significant figures. To decide if a number is a counting number you must relate it to the context of the problem as illustrated in the next sample problem. Sample Problem 11 A new penny weighs 3.8 grams. How much do 7 new pennies weigh? Solution 1. Multiply (3.8) (7) 2. Number of Significant figures 2 unlimited 3. Calculator answer Rounded off to the correct accuracy 27 (to 2 significant figures) 12. Using Scientific Notation to Express the Correct Number of Significant Figures Once in a while we come across a number whose accuracy cannot be expressed correctly as a regular number For example: the number two hundred, if measured on an instrument which measures to the nearest ten, must be expressed as two significant figures. To do so you must express the number in scientific notation as shown below. 200 (one sig fig) 200. (three sig figs) 2.0 x 10 2 (two sig figs) Problems Be sure you study sections 10, 11, and 12 before doing these problems (6.50) (330.) 107. (25) (32) A toothpick weighs 1.45 grams. How much do 70 toothpicks weigh? 110. A sample of metal is measured and found to be inches in length. What is its length in feet? (Twelve inches are "defined" to equal one foot. This is an example of a counting number, not a measurement.) Chap1 Math Concepts-JC.docx 05/12/11 8
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